For a realpositive ,
the Riemann-Siegel function is defined by
(1)
This function is sometimes also called the Hardy function or Hardy -function (Karatsuba and Voronin 1992, Borwein et al. 1999).
The top plot superposes (thick line) on , where is the Riemann zeta
function.
For real ,
the Riemann-Siegel theta function is defined as
(2)
(3)
The function
has local extrema at (OEIS A114865
and A114866).
Values
such that
(4)
for ,
1, ... are known as Gram points (Edwards 2001, pp. 125-126).
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439-462, 1995.Borwein, J. M.; Bradley, D. M.; and Crandall,
R. E. "Computational Strategies for the Riemann Zeta Function." J.
Comput. Appl. Math.121, 247-296, 2000.Brent, R. P.
"On the Zeros of the Riemann Zeta Function in the Critical Strip." Math.
Comput.33, 1361-1372, 1979.Edwards, H. M. Riemann's
Zeta Function. New York: Dover, 2001.Karatsuba, A. A. and
Voronin, S. M. The
Riemann Zeta-Function. Hawthorn, NY: de Gruyter, 1992.Odlyzko,
A. M. "The th Zero of the Riemann Zeta Function and 70 Million of
Its Neighbors." Preprint.Sloane, N. J. A. Sequences
A036282, A114721,
A114865, and A114866
in "The On-Line Encyclopedia of Integer Sequences."Titchmarsh,
E. C. The
Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.van
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Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 143, 1991.